# The throughput of the ALOHA protocol if the Binomial distribution was used

In all the examples of ALOHA I've seen, the Poisson distribution is used. Theoretically, how could the throughput be calculated if a Binomial distribution was used instead?

For example, in the case that the offered load follows a Binomial distribution with probability $p$, I know that the probability of $k$ attempts within $t$ time slots is as follows:

$$\Pr\big[k \text{ attempts within t time slots}\big] = \binom{t}{k}p^k(1-p)^{t-k}\,.$$

I know that the throughput, $S$ is equivalent to $G$ (average offered load) multiplied by the probability of success:

$$S = G\cdot\Pr\big[\text{successful}\big]\,,$$

but what would this be in the case that a binomial distribution is used?

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In contrast, using a binomial distribution doesn't come from any natural model of the individual nodes' behaviour. It only allows you to model "were there at least $n$ transmissions attempts during this time slot?" for some $n$ of your choosing. It doesn't seem able to deal with back-off and it doesn't seem able to deal with multiple collisions within a single time slot.